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- Title
Planar stochastic hyperbolic triangulations.
- Authors
Curien, Nicolas
- Abstract
Pursuing the approach of Angel and Ray (Ann Probab, ) we introduce and study a family of random infinite triangulations of the full-plane that satisfy a natural spatial Markov property. These new random lattices naturally generalize Angel and Schramm's uniform infinite planar triangulation (UIPT) and are hyperbolic in flavor. We prove that they exhibit a sharp exponential volume growth, are non-Liouville, and that the simple random walk on them has positive speed almost surely. We conjecture that these infinite triangulations are the local limits of uniform triangulations whose genus is proportional to the size. Graphical abstract: An artistic representation of a random (3-connected) triangulation of the plane with hyperbolic flavor. [Figure not available: see fulltext.]
- Subjects
TRIANGULATION; MARKOV processes; LATTICE theory; HYPERBOLIC geometry; LIOUVILLE'S theorem
- Publication
Probability Theory & Related Fields, 2016, Vol 165, Issue 3/4, p509
- ISSN
0178-8051
- Publication type
Article
- DOI
10.1007/s00440-015-0638-4